Journal of the Mechanics and Physics of Solids 70 (2014) 136–153
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Coupled glide-climb diffusion-enhanced crystal plasticity M.G.D. Geers a,n, M. Cottura a, B. Appolaire b, E.P. Busso c,d, S. Forest d, A. Villani d a
Eindhoven University of Technology, Department of Mechanical Engineering, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands Laboratoire d'Etude des Microstructures, CNRS/Onera, BP72, 92322 Châtillon Cedex, France c ONERA, BP 80100, 91123 Palaiseau Cedex, France d Mines ParisTech, Centre des Matériaux/CNRS, UMR 7633, BP87, 91003 Evry Cedex, France b
a r t i c l e in f o
abstract
Article history: Received 24 January 2014 Received in revised form 11 April 2014 Accepted 12 May 2014 Available online 21 May 2014
This paper presents a fully coupled glide-climb crystal plasticity model, whereby climb is controlled by the diffusion of vacancies. An extended strain gradient crystal plasticity model is therefore proposed, which incorporates the climbing of dislocations in the governing transport equations. A global–local approach is adopted to separate the scales and assess the influence of local diffusion on the global plasticity problem. The kinematics of the crystal plasticity model is enriched by incorporating the climb kinematics in the crystallographic split of the plastic strain rate tensor. The potential of the fully coupled theory is illustrated by means of two single slip examples that illustrate the interaction between glide and climb in either bypassing a precipitate or destroying a dislocation pile-up. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Crystal plasticity Dislocation climb Vacancy diffusion Dislocation pile-ups Strain gradient
1. Introduction Many metallic systems are nowadays operated in a regime where the evolving mechanical properties do not just depend on dislocation glide mechanisms within the underlying crystals. This is typically the case for climb-assisted and creep deformation mechanisms, which are well known to contribute significantly at higher temperatures in pure metals (Seidman and Balluffi, 1966a,b; Dobson et al., 1967), solid solution alloys (Mohamed and Langdon, 1974), particle strengthened and multi-phase metals (Arzt and Wilkinson, 1986; Beddoes et al., 1995; Koch, 1998). Moreover, in modern thin film systems for micro-electronics or micro-electro-mechanical systems (MEMS), climb mechanisms already contribute to a significant extent at only mildly elevated temperatures (Legoues et al., 1990; Guo et al., 2010; Karanjgaokar et al., 2012) or even room temperature, e.g. in metallic interfaces (Wang et al., 2009; Wang and Pan, 2011). Basic physical concepts of dislocation climb go back to the founding contributions of Hirth and Lothe (Lothe, 1960; Lothe and Hirth, 1967; Hirth and Lothe, 1968; Nix et al., 1971) and the work of Thomson and Balluffi (1962) on a kinetic theory for dislocation climb. Early work on the interaction (or transition) between glide and climb for solid solution alloys (mostly at higher temperatures) can be found in Mohamed and Langdon (1974). Dislocation climb requires particular attention in the study of interfaces, even at room temperature, see e.g. the work of Wang et al. (2009), Wang and Pan (2011), Wang and Misra (2011). The incorporation of atomistic aspects has been addressed by Lau et al. (2009) and Kabir et al. (2010), for BCC Fe within the power-law creep regime. Particular emphasis was placed on vacancy diffusion in FCC metals by Ortega et al.
n
Corresponding author. E-mail address:
[email protected] (M.G.D. Geers). URL: http://www.tue.nl/mechmat (M.G.D. Geers).
http://dx.doi.org/10.1016/j.jmps.2014.05.007 0022-5096/& 2014 Elsevier Ltd. All rights reserved.
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(2002), using a Molecular Dynamics (MD) approach, which might be of interest to determine the diffusion coefficients. A theoretical model in which the climb of dislocations is controlled by vacancy emission from extended jogs has been proposed by Argon and Moffatt (1981). The classical modeling of dislocation climb based on a continuous description of vacancy diffusion has been compared to atomistic simulations of dislocation climb in body-centered cubic iron under vacancy supersaturation by Clouet (2011). To investigate the effect of climb on the collective behavior of dislocations, a number of papers have been published on the coupling of dislocation climb to discrete dislocation dynamics (DDD), e.g. see the work of Raabe (1998), Mordehai et al. (2008) and Ayas et al. (2014). Simplifying assumptions are thereby often made to enable computations to become feasible (e.g. time scale separation by considering a steady-state vacancy flux at each time step; ignoring the elastic interaction energy between dislocations and vacancies; assuming diffusion coefficients to be independent of stress). A DDD-analysis of the effect of climb on dislocation mechanisms and creep rates in superalloys was performed by Haghighat et al. (2013). Furthermore, a dislocation climb model based on coupling the vacancy diffusion theory with 3D discrete dislocation dynamics has been proposed by Gao et al. (2013) and Ayas et al. (2014). Recent DDD-based models coupling glide with climb have been published by Gao et al. (2011a), Keralavarma et al. (2012) and Danas and Deshpande (2013). For thin film problems, coupled DDD models have been proposed and assessed by Ayas et al. (2012) and Davoudi et al. (2012). Recent work has tried to incorporate as much as possible the basic processes occurring at the dislocation cores (Mompiou and Caillard, 2008; Gao et al., 2011b; Danas and Deshpande, 2013; Ayas et al., 2014; Geslin et al., 2014). The role of climb in evolving dislocation patterns has been shown in Bakó et al. (2006), even though the climb problem was solved in a more phenomenological manner (without coupling it to vacancy diffusion). The influence of climb on the back stress (past cuboidal particles) has been discussed by Rö sler (2003), where the back stress was found to be proportional to the applied stress. Xiang and Srolovitz (2006) examined the effects of dislocation climb on bypassing particles, following a DDD approach. To tackle larger length scales which cannot be investigated by atomistic or DDD models, the role of dislocation climb in plasticity has been addressed in several publications in a purely phenomenological manner, where constitutive equations incorporate the contribution of climb, e.g. see references cited in Gao et al. (2011b). The incorporation of climb in crystal plasticity has been dealt with in a phenomenological manner and improved versions thereof by Lebensohn et al. (2012). The main objective of the present work is to break through the phenomenological nature of these approaches at the crystal plasticity level by coupling deformation to dislocation motion assisted by glide and diffusional climb. For this purpose, a state-of-the-art crystal plasticity model needs to be used, which properly incorporates short-range dislocation– dislocation interactions. The backbone of the crystal plasticity framework is based on the strain gradient crystal plasticity model proposed by Evers et al. (2004), Bayley et al. (2006), and Ertürk et al. (2009), enriched with dislocation transport physics (Groma et al., 2003; Yefimov et al., 2004). A second (physical) coupling is made here to link climb with vacancy diffusion. Reference to this is made to the work of Gao and Cocks (2009), which presents a thermodynamically based variational framework that links the climb of a single edge dislocation to the diffusion of vacancies. Attention is given to the governing equations and physics of the vacancy diffusion process, the driving forces acting on a dislocation to climb and the intrinsic coupling between both. The climb law is therefore diffusion-controlled. The innovative contributions of the present paper are: (1) a physically based constitutive framework coupling vacancy diffusion to dislocation climb within a crystal plasticity setting; (2) the incorporation of climbing dislocations in the governing transport equations used in strain gradient crystal plasticity; (3) a global–local approach to separate the scales and assess the influence of the local diffusion problem on the global plasticity problem; and (4) an enrichment of the crystal kinematics, enabling the incorporation of the climb kinematics in the crystallographic split of the plastic strain rate tensor. Section 2 introduces the governing equations for vacancy diffusion and the adopted strain gradient crystal plasticity model. Section 3 presents the coupling between climb and glide, which entails the modified diffusion problem and the modified crystal plasticity problem. Section 4 provides some illustrative single slip case studies highlighting the interaction between diffusion, climb, and glide. Finally, the conclusions are presented in Section 5. 2. Model formulation 2.1. Main physical concepts The interaction between vacancy diffusion and dislocation climb has been studied at the level of individual and discrete dislocations by Gao and Cocks (2009). At this level, henceforth called ‘the local level’, a direct relation exists between the transport of vacancies and the dislocation climb process, involving absorption/emission of vacancies at jogs in the dislocation core. One of the key results is an expression of the local climb velocity as a function of the associated quantities, i.e. the local concentration gradient and the mean concentration along the dislocation core. This mean concentration depends on the configurational forces acting on the dislocation (see e.g. Gao and Cocks, 2009; Danas and Deshpande, 2013), composed of the following contributions: (1) the so-called osmotic force, which results from the change of free energy upon creating or removing vacancies during dislocation climb; (2) the drag force acting on the dislocation, which results from the change of free energy of vacancies due to the pressure evolution during dislocation climb (see Section 3.1); and (3) the climb component Fc of the Peach–Koehler force.
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The Peach–Koehler force is given by ! ! ! F PK ¼ b ðr m Þ p ð1Þ ! ! ! ! ! with b the Burgers vector length, m the glide direction, n the slip plane normal and p ¼ m n . The tensor r is the total Cauchy stress tensor, i.e. the sum of the stress due to the external load and the internal stress due to e.g. short range ! dislocation interactions. The component contributing to climb (acting along the direction n ) then reads ! ! ! ! ! F c ¼ F PK n ¼ b ½ðr m Þ p n ð2Þ We will study the problem at the meso-scale, where pile-ups against particles can be resolved explicitly so that it is meaningful to establish a direct coupling between the mean vacancy flux and the dislocation climb process. We limit ourselves to 2D problems governed by edge dislocations only. At that scale, the total amount of dislocations is split in a positive fraction ρ þ and a negative fraction ρ . The sign of the dislocation results from the position of the extra half-plane with respect to the slip plane normal, or alternatively, with the orientation of the Burgers vector with respect to the slip orientation vector. The difference between ρ þ and ρ gives the density of the geometrically necessary dislocations ρG ¼ ρ þ ρ (GNDs) and the total dislocation density ρ, which is the sum ρ ¼ ρ þ þ ρ . The dislocation problem can be described in a fully conservative manner with transport equations. Since the analysis is carried out at the meso-scale, all dislocations are assumed to participate in the climbing process. The corresponding dislocation transport equations have to be updated to incorporate the contribution through climbing. 2.2. Governing equations for vacancy diffusion The thermodynamical setting for the vacancy diffusion problem is typical for a standard diffusion problem. This subsection focuses on the governing equations for the diffusion problem. The governing quantities and field variables, along with their notations, are
vacancy concentration ¼vacancy volume fraction, c [–] ! volumetric vacancy flux, J v [m/s] mobility, M [m5/ Js] vacancy diffusion constant, D [m2/s] molar volume, Vm [m3/mol] relaxed volume per mole of vacancies, ΔV r [m3/mol] molar vacancy formation energy, Qv [J/mol] diffusion potential, μv [J/m3] or [N/m2] or [Pa] absolute temperature, T [K] universal gas constant, R [J/(K mol)]
For a definition of the diffusion potential, here denoted conservation of mass, or continuity equation, reads -
μv,
refer to the work of Larché and Cahn (1985). The local
-
c_ ¼ ∇ J v
ð3Þ
A linear relation is classically postulated to hold between the vacancy flux and the driving force for diffusion: ! J v ¼ M ∇ μv
-
in V
ð4Þ
The volumetric diffusional mobility M is related to the vacancy diffusion constant D (Gao and Cocks, 2009) by M¼
DV m c ð1 cÞ RT
ð5Þ
Within a thermodynamical setting, assuming that vacancies constitute an ideal solid solution, the diffusion potential can be determined (e.g. Larché and Cahn, 1985; Gao and Cocks, 2009) as RT c Q 1trðrÞΔV r þ v in V ð6Þ μv ¼ ln Vm 1 c RT 3 RT with the following boundary conditions (on the external boundary ∂V) for the diffusion potential μv (Gao and Cocks, 2009) ! ! μv ¼ n r n on ∂V ð7Þ The origin of the different terms in (6) is
RT=V m ln ðc=ð1 cÞÞ: entropy of mixing of the vacancies in the lattice Qv: vacancy formation energy, i.e. the enthalpy change associated with the breaking of actual bonds in the region of the vacancy under zero local pressure (constant for a given crystal)
13trðrÞΔV r : elastic energy released when an atom is removed locally, which is the negative work done by the local pressure
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Particular emphasis has to be given to the coupling of vacancy diffusion with dislocation climb, i.e. how that reversely affects the transport of vacancies. 2.3. Governing crystal plasticity equations A rate-dependent crystal plasticity formulation is adopted, where dislocations are assumed to be mobile during deformation. The problem will be studied in the adopted 2D setting, in which only edge dislocations are considered. For the present work, a small strain formulation is adopted. The classical governing equations for the glide only case are summarized next:
The additive decomposition of the infinitesimal strain tensor ε into its elastic part εe and its plastic part εp ε ¼ εe þ εp
ð8Þ
The constitutive equation, coupling the Cauchy stress tensor r to the elastic strain tensor εe through the FCC-anisotropic elastic fourth-order constitutive tensor 4 C
r ¼ 4 C : εe
ð9Þ
The crystallographic split, coupling the plastic strain rate tensor ε_ p to the crystallographic slip rates γ_ α and the !α !α !α !α !α !α symmetrized Schmid tensor Pα ¼ 12 m n þ n m , with m and n the slip direction and slip plane normal, respectively
ε_ p ¼ ∑ γ_ α Pα α
ð10Þ
When focusing on a single slip system only, the superscript α will be omitted. Rewriting the scalar climb component of the Peach–Koehler force (2) as a projection on the slip system gives ! ! ! ð11Þ F c ¼ b ½ðr m Þ p n ! ! ! ! ! F c ¼ b ðr m Þ ð p n Þ ¼ b ðr m Þ ð m Þ
ð12Þ
! ! Fc ¼ b r : m m
ð13Þ
We will next focus on the slip law defining γ_ and the dislocation transport equations. A viscous drag relation is applied between the average dislocation velocity field v and the driving force Fg acting on the dislocations (i.e. the glide component of the Peach–Koehler force): vðx; t Þ ¼
Fg B
ð14Þ
where the parameter B is the drag coefficient. The driving force acting on a dislocation consists of an elastic component emanating from the long range elastic fields and an internal stress component resulting from short-range dislocation interactions. Thus, F g ðx; tÞ ¼ b ½se þ si :
ð15Þ
In the above equation, the stress se is the projection of the external stress vector (acting on the glide plane) on the glide direction, i.e. the externally resolved shear stress ! ! se ¼ ½ðre ! mÞ p m ð16Þ ! se ¼ ! n re m
ð17Þ
The external stress tensor re corresponds to the stress defined in Eq. (9), whereby only the long range elastic fields are considered in εe . The internal stress si results from the short range discrete interactions between dislocations, i.e. through a gradient in the GND density. The internal stress here reads G b ∂ρG G b ∂ρ þ ∂ρ ¼ ð18Þ si ¼ 2π½1 ν ρ ∂x 2π½1 ν ρ ∂x ∂x whereby G is the shear modulus and ν Poisson's ratio. Note that this shear internal stress si does not contribute to the climb component of the Peach–Koehler force Fc, since it cancels out in the projection used in Eq. (13). Eq. (18) was developed by Groma et al. (2003) and Yefimov et al., (2004), using statistical mechanics arguments applied to nearest neighbor interactions. Note also that similar dislocation interaction models exist, see Evers et al. (2004), Bayley et al. (2006), and Ertürk et al. (2009).
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For simplicity, the classical Orowan relation for the slip rate γ_ is here adopted, γ_ ¼ ρbv
ð19Þ
The balance equations for the glide-only problem are given by (i) GND balance equation ∂ρG ∂ þ ðρvÞ ¼ 0 ∂t ∂x
3
∂ρ þ ∂ρ ∂ ρþ þρ v ¼ 0 þ ∂x ∂t ∂t
ð20Þ
This equation implies the conservation of the total Burgers vector (as can be verified by an integration over the entire domain). (ii) balance of the total density of dislocations: ∂ρ ∂ þ ρ v ¼s ∂t ∂x G
3
∂ρ ∂ þ ρ þ ρ v ¼ s ∂t ∂x
ð21Þ
This second balance equation is a transport equation. It was proposed by Groma et al. (2003), using statistical mechanics arguments (for an extensive discussion on this second balance equation, see also Hirschberger et al., 2011). In the absence of a source or sink, i.e. s ¼0, Eq. (21) implies the local conservation of the total number of dislocations. Details on the role of the convective contribution in this term are discussed in Hirschberger et al. (2011). The coupled problem consisting of the two partial differential equations (20) and (21) describes the evolutionary behavior of all dislocations. The source and sink of dislocations are described by the source term s on the right-hand side of Eq. (21). 3. Coupled glide-climb crystal plasticity 3.1. Modified diffusion problem In general, one has to distinguish two distinct limit regimes for the coupling between bulk vacancy diffusion and dislocation climb. To clarify this, the process at the core and dislocation line needs to be detailed. Among the three configurational forces expanding power during a dislocation climb process, mentioned in Section 2.1, the drag force induced by the elastic interaction between the dislocation and the vacancies can usually be neglected. Indeed, the contribution of trðrÞΔV r is much smaller than the work performed by the climb component of the Peach–Koehler force FPK (see Gao and Cocks, 2009). Therefore, the last term in the diffusion potential μv in Eq. (6) can be dropped. The osmotic force per unit length of an edge dislocation Fo is directly related to the diffusion potential μv and equals bμv . Using Eq. (6) in the dilute limit, i.e. cv 5 1, the osmotic force Fo is given by the expression proposed by Hirth and Lothe (1968) ⋆ RT b c Fo ¼ ln ð22Þ Vm c0 Here, c⋆ represents the vacancy concentration in the core of the dislocation, whereas c0 ¼ expð Q v =RTÞ is the standardstate concentration of vacancies (i.e. the equilibrium concentration at temperature T if no dislocations are present). Neglecting the drag force, the total force that expands power when a dislocation climbs is then given by ⋆ RT b c F ¼ Fc þ Fo ¼ Fc þ ln Z0 ð23Þ Vm c0 When F¼0, the dislocation core is at local equilibrium and F cV m c⋆ ¼ ceq ¼ co exp bRT
ð24Þ
In that case, vacancy dynamics (creation and destruction) is fast at jogs and the equilibrium concentration ceq is therefore easily maintained at a jog. The concentration at the core c⋆ depends on the jog content and spreading. If the mean free path of the vacancies along the dislocation line is larger than the jog spacing, then the whole dislocation line will have a vacancy concentration nearly equal to ceq (and F¼0). In such case, the rate of vacancy absorption is then only limited by the vacancy bulk diffusion rate towards the dislocation. This is the diffusion-limited regime. For this regime, the dislocations locally maintain the equilibrium concentration ceq in the core, which is directly related to the climb component of the Peach– Koehler force acting on it. If the mean free path is smaller than the jog spacing (either because diffusion along the core is slow or there are few jogs), then the rate of vacancy absorption is limited by the rate at which vacancies are absorbed at the jogs along the dislocation line. This is the sink-limited regime. In this regime, an expression relating the driving force F and the climb velocity must be provided on the basis of the elementary processes acting at the dislocation core. By eliminating c0 in Eq. (23) by making use of Eq. (24), the following expression for the driving force results F ¼ RTb=V m lnðc⋆ =ceq Þ. As usually done in irreversible thermodynamics, if F is not too large a linear relationship between the climb velocity vc and F can be postulated, i.e. vc ¼ K c F where Kc is a positive kinetic coefficient. This amounts to add to the dissipation potential Ψ in Gao and Cocks (2009) a term quadratic in the climb velocity. It is worth noting that if c⋆ is close to ceq, F can be linearized with respect to the concentrations as F ¼ RTb=V m ðc⋆ ceq Þ. Then, the climb velocity and the flow of vacancies are proportional to
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c* = ceq c
c Fig. 1. Global and local scales for diffusional climb.
c⋆ ceq as in Ayas et al. (2012) and Danas and Deshpande (2013). The extension of the present model to the sink-limited regime in this manner is thus straightforward, since it changes only the climb velocity at the discrete dislocation level, Eq. (33). The required additional equation for c⋆ can be easily obtained following the procedure proposed by Danas and Deshpande (2013) and Ayas et al. (2012), see e.g. Eq. (13) in Danas and Deshpande (2013) (in which c0 is here c and in which c is here c⋆ ). Depending on the material properties controlling the processes at the dislocation cores (diffusion, jog nucleation, vacancy emission/absorption), one of the two regimes mentioned can be considered as operating during a whole process. These properties are not always trivial quantities to identify. Any mechanism that promotes the diffusion along the core or the dislocation line will favor the diffusion-limited regime. Since pipe diffusion is known to be significant in many cases, we consider the diffusion-limited regime as the most relevant one here. A local and a global scale need to be distinguished next. The local scale is that of the discrete dislocation, for which we will denote the vacancy concentration field in its neighborhood simply by c. The global scale corresponds to that of a macroscopic material point with its connected unit cell or representative volume element (RVE). The RVE is the representative material volume within which local dislocations reside, represented by a dislocation density ρ, see Fig. 1. The average vacancy concentration in that RVE is denoted c. The transport of vacancies across the RVE boundaries has to be solved by a global diffusion equation applied to the global c, reflecting the variations in the mean vacancy concentration between neighboring RVEs. Note that mean-field assumptions only apply to the global scale, since the solution of the local scale problem will be done analytically. In summary, it is assumed that
Macroscopic quantities associated with the material point are considered constant with respect to the RVE. Dislocations with a given sign within the RVE are equivalent. They are exposed to the same macroscopic fields and the local vacancy concentrations surrounding them are identical.
All local problems related to individual dislocations in the RVE are identical, as a result of the mean field assumption adopted. Hence, all positive dislocations will have the same climb velocity. Likewise for the negative dislocations, where the climb velocity will be equal and opposite to the positive ones.
3.1.1. The local problem The local problem aims to determine the amount of vacancies consumed or emitted by all the dislocations, which should be used as a source term in the global problem. At the level of a single positive dislocation, whereby the core is enclosed within Γ (a cylindrical surface of unit length, ! with a unit normal vector N pointing inwards towards the core), the coupling with the local climb velocity vc [m/s] can be expressed as I 1 - ! vc ¼ J N dΓ ð25Þ b Γ v which needs to be solved to extract the effective climb velocity resulting from the diffusion of vacancies into the dislocation core. Note that the dislocation line length considered in Eq. (25) is a unit length. Eq. (25) also shows that the core of a positive dislocation that climbs upwards (positive vc) is a vacancy sink. Likewise, a negative dislocation is a vacancy source if it climbs upwards. Consider a small cylindrical volume B at the local scale, and a single slip system with crystallographic unit directional !!! vectors n ; m ; p . The dislocation density ρ þ is the total line length of positive edge dislocations per unit volume. The total dislocation line length in the volume B is L. The climbing dislocations in volume B are considered as point sources or sinks. Using a continuum setting and considering all climbing dislocations as points sources or sinks allows us to apply Eq. (25) to the considered volume B. Eq. (25) reveals the inward flux into the core, i.e. a positive flux implies a decrease of the vacancy concentration in the vicinity of the core. Considering the climb velocity as a field property (i.e. constant in B) then
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leads to Z B
c_ dB ¼ L vc b
ð26Þ
Recognizing that the sum of all the dislocation line lengths (of all dislocations in B) is the integrals of ρ þ dB over the volume B, allows us to rewrite Eq. (26) as Z Z ð27Þ c_ dB ¼ ρ þ vc b dB B
B
! Here, the climb velocity, vc, acts along the normal direction n . Evidently, the dislocation density ρ þ in Eq. (27) corresponds to that of the climbing ones. Within the adopted mean field approach, all dislocations in the volume contribute to climb. Since Eq. (27) has to hold for any choice of B, the following continuum coupling equation for the local concentration of vacancies results c_ ¼ ρ þ vc b
ð28Þ
We now incorporate both positive and negative dislocations (each with positive densities ρ þ and ρ , respectively), since their climbing velocities have opposite sign under the influence of the same climb force. Hence, Eq. (28) reads c_ ¼ ðρ þ vcþ ρ vc Þ b vcþ
ð29Þ
vc
with and the climb velocities of the positive and negative dislocations, respectively. Eq. (29) clearly shows that a positive dislocation climbing up under the influence of a compressive smm stress (with a positive climb velocity) consumes vacancies, through which the local vacancy concentration will decrease, thus resulting in a negative c_ . Likewise, a negative dislocation in the same volume subjected to a compressive smm stress will climb down with a negative velocity, contributing equally to a decrease in the local vacancy concentration. The local evolution of c enters the global scale through a source term Sv given by the right-hand side of Eq. (29), depending on the net dislocation flux ρ þ vcþ ρ vc , that has entered (or left) the global volume element through climb. This source term reads Sv ¼ ðρ þ vcþ ρ vc Þ b
ð30Þ
As emphasized later in the text, under the influence of the same stress (and hence climb force), negative dislocations will always experience a climb velocity that is equal and opposite to that of positive dislocations. This consideration holds within the considered volume element representing a single material point. Hence, Eq. (30) can also be formulated as Sv ¼ ρ vcþ b
ð31Þ
3.1.2. The global problem The flux of climbing dislocations (positive or negative) ρ þ vcþ ρ vc appearing in the source term Sv at the global level depends on the rate of change c_ and the local equilibrium concentration at the level of the discrete dislocations: ρvcþ ¼ ρ þ vcþ ρ vc ¼ f ðc ̇ ; ceq Þ
ð32Þ
If the global vacancy concentration c equals the local equilibrium concentration ceq (given by Eq. (24)), then climb should stop. Based on the solution of a stationary problem in a hollow cylinder, Bakó et al. (2011) proposed an expression for the climb velocity of dislocations, precisely satisfying this requirement. The climb velocity is determined at the level of the discrete dislocations, yielding
2 π D c0 ceq c vc ¼ ð33Þ b lnðr 1 =r c Þ c0 c0 Here, the radius r 1 is the distance at which it can be assumed that the local concentration c equals c. A typical value for r 1 is pffiffiffi half the dislocation spacing, i.e. r 1 ¼ 1=ð2 ρÞ. The core radius rc is typically taken as r c ¼ 4b. The sign of the climb velocity in the climb direction depends on the dislocation sign and on the sign of the climb force acting on the dislocation. Note that the climb velocity is coupled to the stress state through ceq via Eq. (24). It should be recalled that the climb contribution of ! the Peach–Koehler force under normal loading in the glide direction m equals F c ¼ b smm . To illustrate the possible climb configurations that a pair of dislocations, either positive or negative, may adopt, the climb directions and evolution of corresponding vacancy concentrations are schematically shown in Fig. 2. ! Under compression in the m direction, smm is negative. Hence, positive dislocations with a climb direction that is ! ! positively oriented with respect to m will have a positive climb velocity (upward climb, in the direction of n ). Likewise, ! negative dislocations with a climb direction that is negatively oriented with respect to m will have a negative climb velocity (downward climb). If smm reverses sign, then the climb velocities will switch sign as well, and so will the climbing fluxes ρ þ vcþ and ρ vc .
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c c
c c
c
c
c
c
c c
c
c c
c
c
c
c
c
Fig. 2. Climb configurations under the influence of a normal stress smm .
t
dY
dX
dy
dx
Fig. 3. Positive dislocation climbing over a distance dY in a volume dV¼ dX dY during a time step Δt.
As climbing dislocations consume or emit vacancies, this process should enter the global diffusion problem through a source term, i.e. ! ! c_ ¼ ∇ J v þ Sv
ð34Þ
Here, Sv results from the local scale, including all the individual contributions of all dislocations in the volume element representing a global macroscopic point. This source term was expressed in Eq. (30) in terms of the positive and negative dislocations. With dislocation densities ρ þ , ρ , as well as the climb velocities known, Eq. (34) is now fully defined.
3.2. Modified crystal plasticity formulation accounting for dislocation climb 3.2.1. Climb contribution to the kinematic fields In order to incorporate the effect of the dislocation climb on the plastic deformation tensor, the crystallographic split of the plastic strain rate tensor needs to be extended for the climb kinematics associated to each slip system. The crystallographic split for ε_ p used in Eq. (10) needs to be extended to account for the climb kinematics. Then, α
ε_ p ¼ ∑ γ_ α Pα þ ∑ β_ Q α α
α
ð35Þ
in which the climb projection tensor Q α related to each slip system α reads !α !α Qα ¼ m m
ð36Þ
α We next determine β_ on the basis of the simple kinematics of a climbing dislocation on a single glide plane α, as represented in Fig. 3. The superscript α is henceforth omitted for brevity. The height of the volume before and after climb is unaffected, i.e. dy ¼dY. The rate of change of the local vacancy concentration c_ is given by Eq. (28), which is evaluated over the time step Δt for the dislocation to climb over the vertical distance dY:
c_ ¼ ρ þ vcþ b Δc ¼ ρ þ ðvcþ ΔtÞ b
ð37Þ ð38Þ
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The volume occupied by the change in vacancies is denoted ΔV and directly results from the scalar change of the vacancy concentration Δc. Then, ΔV ¼ Δc dX dY
ð39Þ
The volume change ΔV can also be obtained from the deformed configuration ΔV ¼ dx dy dX dY. When combined with Eq. (39) dx dy dX dY ¼ Δc dX dY
ð40Þ
ðdx dXÞ dY ¼ Δc dX dY
ð41Þ
dx dX ¼ Δc dX
ð42Þ
β ¼ εxx ¼ Δc
ð43Þ
One finds, as expected, that the only kinematical contribution of the climbing dislocation is a normal strain εxx that exactly _ equals the local variation of the vacancy concentration. In a rate form, Eq. (43) gives the expected result for β, β_ ¼ c_ ¼ ð ρ þ vcþ ρ vc Þ b
ð44Þ
which can be directly used in the crystallographic split equation (35), whereby the solution of the local problem gives the flux of climbing dislocations. 3.2.2. Extended dislocation transport The transport of dislocations through climb naturally modifies the balance equations (20) and (21). Taking into account the flux of positive and negative climbing dislocations, requires the balance equation for the total amount of dislocations to be extended. To clarify this point, we consider adjacent volume elements in Fig. 4 with a boundary through which the climbing flux may impact the vertical (climbing direction) redistributions of dislocations. Without any gradient in the densities of the positive (Fig. 4a) or negative (Fig. 4b) dislocations, an equal amount of dislocations leaving the top (or bottom) boundary will enter on the opposite bottom (or top) boundary. If so, the climb processes will affect the redistribution of vacancies but it will not on average change the dislocation densities. This is clearly different if a gradient exists along the climb direction for either the positive or the negative dislocations, as schematically depicted in the figure. The balance equation of the total amount of dislocations in a volume element, Eq. (21), is directly affected through these additional transport terms. The divergence of the flux of climbing dislocations should be added to Eq. (21), which for the single slip problem is limited to the gradient with respect to the vertical (climb) direction y. The resulting equation reads ∂ ∂ρ ∂ þ ρþ ρ v þ ρ v þ þρ vc ¼ s ∂t ∂x ∂y þ c
ð45Þ
This balance equation follows directly from the inflow and outflow of dislocations in the considered central volume element. Recall that within the same volume element, subjected to the same stress (and hence climb force), the negative dislocations will always have a climb velocity that is equal and opposite to the climb velocity of the positive dislocations. This allows Eq. (45) to be rewritten as follows: ∂ ∂ρ ∂ þ ρþ ρ v þ ρ þ ρ vcþ ¼ s ∂t ∂x ∂y
ð46Þ
or expressed in terms of the density of geometrically necessary dislocations ρG ∂ρ ∂ ∂ þ ρ v þ ρ vþ ¼ s ∂t ∂x G ∂y G c
ð47Þ
4. Single slip case studies To illustrate the coupled framework and the interaction between glide and climb, we will present two 2D examples with a single slip system. The single glide plane case optimally focuses on the interaction between glide and climb (Ayas et al., 2012), since dislocations cannot glide past obstacles in 2D without climbing. Evidently, these results do not address the more complex physical behavior of climb for multiple glide systems but it does provide the best illustration of the proposed theory. The glide-climb problem considered is fully conservative, i.e. no sources or sinks for dislocations are incorporated and transport is therefore assumed to be dominant. Since the relaxation of the elastic waves is by orders of magnitude faster than the evolution of plastic strain resulting from dislocation motion, static mechanical equilibrium will be assumed. At any time, the elastic strain can be computed by solving mechanical equilibrium at constant plastic strain. This is done iteratively using a fix point algorithm. A spectral method based on Fast Fourier Transforms (FFT) (Moulinec and Suquet, 1998) is employed to solve the governing equations.
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ceq
145
ceq c
c
Fig. 4. Dislocation redistribution through climb in the presence of a vertical dislocation gradient. (a) Climb up, (b) climb down.
Table 1 Model parameters. b (m)
B (Pa s)
D (m2 s 1)
c0
G (GPa)
ν
2.5 10 10
10 3
10 6
10 4
26.3
0.33
The dislocation densities and vacancy concentration evolution as well as the plasticity part of the problem are solved using an explicit time-stepping scheme. In addition, the convection-dominated problem represented by the dislocation density evolution equations (47) and (20) is solved with a first-order upwind scheme. 4.1. Model parameters The material parameters adopted in the examples representative for the materials of interest (e.g. aluminum alloys) are presented in Table 1. Isotropic elastic moduli have been chosen for simplicity, since the effect of elastic inhomogeneities and anisotropy for these materials does not impact the presented results qualitatively. The value for the vacancy diffusivity has been chosen to illustrate the effect of dislocation climb relative to glide, inducing a climb rate that is three orders of magnitude smaller than the glide rate. This value is taken for numerical expediency only, whereby other values that are orders of magnitude lower give similar results in the considered configurations. Incorporating a realistic diffusivity, with a
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dislocation climb rate that may be ten orders of magnitude smaller than the slip rate would require an adaptive scheme that allows for a distinct time step for climb and glide, as e.g. done in the work of Keralavarma et al. (2012). All calculations have been performed with 2562 nodes. It has been verified that the results are converged with respect to this spatial discretization because the differences are less than 0.1% compared with calculations carried out with 5122 nodes.
4.2. Case study 1: elastic precipitate inside a plastic matrix The first example aims to reveal the influence of climb on glide in the presence of a hard impenetrable precipitate. The addition of climb enables dislocations to escape from the pile-up by climbing over the precipitate. A periodic configuration of a circular elastic precipitate inside a plastic matrix is considered, see Fig. 5. The two-dimensional domain size is 4 4 μm2 . The precipitate with diameter D ¼ 1:2 μm is positioned at the center of the domain. A phenomenological phase ! field variable φð r Þ is used to describe the two-phase microstructure (with φ ¼ 0 in the matrix, φ ¼ 1 in the precipitate). In order to avoid numerical fluctuations, interfaces are smoothed over a width of 4 nodes. The precipitate is fully elastic with the same elastic moduli as its surrounding matrix (G and ν values are given in Table 1). The initial dislocations are homogeneously distributed in a horizontal band impinging the precipitate (Zone 1 in Fig. 5), whereas the rest of the matrix is initially dislocation-free (Zone 2 in Fig. 5). The width of Zone 1 equals 0:75 μm, thus assumed to be smaller than the particle diameter. If this width were either equal or larger than the particle diameter, then dislocations would be able to bypass the particle through glide only in a trivial manner, which would be of no interest to the present study. The initial dislocation density in the horizontal band equals ρ0 ¼ 5 1012 m 2 , yielding an average total dislocation density of ρ ¼ 7:5 1011 m 2 for the entire domain. The total amount of dislocations is conserved, since sources and sinks are omitted in this analysis. Plasticity inside the matrix results from dislocation motion on one slip system oriented parallel to the x1direction. Periodic boundary conditions are used to solve the coupled problem. A global tensile shear deformation of ε 12 ¼ 10 3 is applied and maintained constant (relaxation test). The prediction for the glide-only case will next be confronted with the climb-assisted glide case. The evolution of the ! ! geometrically necessary dislocation density field ρG ð r Þ and the plastic shear εp12 ð r Þ is presented in Figs. 6 and 7, respectively. At t¼0 (not shown), there are no GND dislocations, since positive and negative dislocations are initially distributed homogeneously inside the horizontal band. Upon applying the load, positive dislocations move to the right while negative dislocations go to the left. Dislocations glide until they form pile-ups at the precipitate walls, as can be seen in the GND density distribution in Fig. 6 at t ¼ 0:5 μs. In the matrix Zone 2, there were no initial dislocations. At the onset of deformation, when glide is still the principal mechanism of dislocation motion, the dislocations in the two case studies exhibit similar behavior and reach comparable spatial distributions. After a certain time, the GND distribution does not evolve anymore since dislocation glide is restricted to the respective slip planes in the glide-only case. An equilibrium state is reached whereby dislocations are confined in Zone 1 against the precipitate (Fig. 6 – top). However, when dislocation climb is accounted for, the positive (respectively negative) dislocations piled-up on the upper half of the left (respectively right) wall of the precipitate and move up while the ones on the lower half move down (Fig. 6 – bottom). The GND density decreases, gradually destroying the pile-ups over time. Positive and negative dislocations are relocating in the same area above and below the precipitate. The role of dislocation climb becomes even more apparent upon analyzing the evolution of the plastic shear strain field εp12 ð! r Þ, see Fig. 7. The previously described stages in dislocation motion explain the evolution observed. In the first stages after loading, the splitting of the positive and negative dislocations to the left and right sides of the precipitate walls generates a plastic shear strain in Zone 1 in both the glide-only and the climb-assisted glide case (Fig. 7). Since there are no ! ! initial dislocations in Zone 2, εp12 ð r Þ remains equal to zero. Once the pile-ups have formed, εp12 ð r Þ does not evolve anymore
2 1 2
Fig. 5. Configuration considered in Case 1; the precipitate is black and initial dislocations (ρ ¼ ρ þ ) are located in the indicated band (Zone 1) only; Zone 2 is initially dislocation-free.
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! Fig. 6. Evolution of ρG ð r Þ at different times t ¼0.5, 2.0, 6.0 and 10:0 μs for the glide-only case (top) and the climb-assisted glide case (bottom). Note that there is no evolution anymore for the glide-only case, since dislocations cannot escape once the pile-ups have been formed.
Fig. 7. Evolution of
εp12 ð! r Þ at different times (0.5, 2.0, 6.0 and 10:0 μs) for the glide-only case (top) and the climb-assisted glide case (bottom).
in the glide-only case. In the climb-assisted glide situation, the gradual destruction of the pile-ups entails dislocations climbing upward and downward, producing a plastic strain around the precipitate until they are able to glide again around the curved interface. Dislocations that have climbed past the precipitate enter Zone 2, where they are remobilized to glide, ! leading to an increase in εp12 ð r Þ with time under the applied deformation. The net result is a slightly higher plastic deformation close to the precipitate, and the formation of two parallel slip bands. Note that the plastic region near the particle in Zone 2 results from the fact that, once dislocations climb out of the pile-up, they will interact with other
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dislocations and experience a different interaction stress, si. As the resulting overall internal stress is reduced, dislocations therefore glide back slightly (away from the particle), which leads to a plastically deformed region of finite width around the particle in Zone 2.
! Fig. 8. Evolution of c c0 ð r Þ at different times t¼ 0.5, 2.0, 6.0 and 10.0 μs for the climb-assisted glide case.
! ! Fig. 9. Evolution of s11 ð r Þ and εp11 ð r Þ at different times t ¼0.5, 2.0, 6.0 and 10.0 μs for the climb-assisted glide case.
p ! Fig. 10. Map of s11 ð r Þ at instant t ¼ 10:0 μs for the climb-assisted glide case without the e11 kinematical enrichment in the crystal plasticity framework.
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149
Fig. 11. Relaxation of the resulting global shear stress s 12 over time at constant applied shear: comparison of the glide-only case with the climb case.
In the climb-assisted glide case, climbing dislocations move towards slip planes in Zone 2, which is accompanied by ! diffusing vacancy fluxes. In Fig. 8, the difference between the local concentration of vacancies c ð r Þ and the standard state concentration c0 is presented. In the upper left quarter of the simulation domain, there is a shortage of vacancies while there is a supersaturation in the lower left quarter. Positive dislocations situated at the left of the precipitate in Zone 1 are able to climb upward and downward around the precipitate to reach Zone 2. The resulting non-equilibrium vacancy concentration emerges from the distribution of the normal stress s11, i.e. the climb driving force. The spatially heterogeneous distribution of the normal stress s11 is due to the microstructure of the two-phase material with an elastic inclusion. Positive (respectively negative) dislocation pile-ups against the impenetrable elastic walls of the precipitate generate a s11 stress component that is compressive (respectively tensile) at the top and tensile (respectively compressive) at the bottom of the pile-up (Fig. 9 – top). The spreading of the pile-ups due to dislocation climb is accompanied by the evolution of s11 in Fig. 9, giving rise to a local normal stress peak at the top and bottom of the precipitate. The role of the extended climb kinematics, addressed in Section 3.2.1, is next assessed. As gliding dislocations generate a shear plastic deformation, climbing dislocations produce a normal strain εp11 due to the local variations of the vacancy concentration field. Positive dislocations climbing up introduce a tensile normal strain, as shown in Fig. 9. This kinematic contribution accompanying dislocation climb relaxes the local stresses, as may be expected. To emphasize this, the normal stress field s11 for the case where the kinematic climb extension in Eq. (35) was not accounted for is shown in Fig. 10. Obviously, the stress levels reached are much higher compared to those for the climb kinematic-enriched case shown in Fig. 9 (top right). Even though the dislocation pattern obtained is not significantly affected for this particular example (not shown), the contribution of climb to the plastic strain has a strong influence on s11 and therefore on the climb kinetics. Since the imposed shear strain ε 12 (see Fig. 5) is maintained during the entire test, the resulting global shear stress s 12 will relax over time. This is shown in Fig. 11, where the glide-only case is compared with the glide-climb case. As can be expected, the relaxation of the global shear stress s 12 is significantly enhanced by climb. The incorporation of the normal strain in the climb kinematics for the crystal plasticity model is essential, and contributes largely to the observed relaxation.
4.3. Case study 2: 2D walls The previous example focused on the role of an elastic precipitate and the re-activation of glide once dislocations bypass the precipitate by climbing. In this second example, emphasis is given on a case where re-activation of glide is not possible and dislocation redistribution is the main outcome. This example convincingly shows how climb may have an important role in the relaxation of dislocation pile-ups. For this purpose, a 2D configuration has been chosen, which represents a periodic arrangement of elastic precipitate layers and plastic matrix channels along the x1 direction (Fig. 12). The twodimensional system's physical size is ‘2 ¼ 2 2 μm2 (again discretized with 2562 nodes) with precipitate layers of 0:1‘. The precipitate interface acts as a hard barrier for dislocations and therefore remains elastic. The elastic properties are homogeneous throughout the domain, with Young's modulus E and Poisson's ratio ν given in Table 1. The initial dislocations in the system are stored in two horizontal bands of thickness 0:16‘ and separated by ‘=2 (Fig. 12). The initial total dislocation density is homogeneously distributed in these bands and equals ρ0 ¼ 1013 m 2 . With respect to the entire volume, this corresponds to an average total dislocation density of ρ ¼ 2:9 1012 m 2 . As before, dislocation nucleation and annihilation are not incorporated and dislocations are relocated by transport only (s ¼0). The slip system orientation and boundary conditions are consistent with the previous example. ! The spatial distribution of the GND density ρG ð r Þ at different times is presented in Fig. 13 for the glide-only and the climb-assisted glide case. Under the applied tensile shear strain, dislocations form pile-ups against the elastic walls with positive (resp. negative) dislocations on the right (resp. on the left). The length of the pile-ups is proportional to the shear
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Fig. 12. Pile-up of dislocations walls.
! Fig. 13. Evolution of ρG ð r Þ at different times t ¼0.2, 4.8, 8.0 and 16.0 μs for the glide-only case (top) and the climb-assisted glide case (bottom).
Fig. 14. Maps of
s11, c c0 and the total flux ρ þ vcþ þ ρ vc ¼ ρG vcþ
at t ¼ 0:3 μs for the climb-assisted glide case.
stress and the average total dislocation density. After the second instant shown in Fig. 13 (top), the microstructure does not evolve anymore in the glide-only situation. In the climb-assisted glide, the pile-ups are spreading out vertically: dislocations at the top of the band climb upward while the ones at the bottom climb downward (Fig. 13 – bottom). The system gradually evolves to an infinite low energy wall.
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The climb motion of the dislocations is again governed by the s11 distribution and the accompanying vacancy diffusion. The normal stress s11 at t¼0.3 μs is represented in Fig. 14 (left). The finite pile-ups of positive (resp. negative) dislocations against the 2D impenetrable elastic walls produce a compressive (resp. tensile) normal stress at their top and inversely at their bottom. It leads to a vacancy depletion (resp. supersaturation) at the top of the pile-ups (Fig. 14 center). Therefore, the positive dislocations on the upper half of the pile-up climb upward by absorbing vacancies, triggering vacancies in the neighborhood to diffuse towards the existing gap; the positive dislocations on the lower half of the pile-up climb downward by emitting vacancies which diffuse away from the supersaturated pit. This is consistent with the dislocation fluxes in Eq. (30), presented in Fig. 14 (right). ! In Fig. 15, the evolution of the plastic shear strain εp12 ð r Þ for both the glide-only (top) and the climb-assisted glide (bottom) cases is presented. The positive and negative dislocations homogeneously distributed in the two bands generate a positive plastic shear strain in these areas when gliding to reach their respective walls in the first stages of the simulation. ! The glide-only case saturates once the finite walls are formed. For the climb-assisted case, climb significantly affects εp12 ð r Þ, whereby climb always entails subsequent glide towards the elastic wall. A negative plastic shear appears, resulting from dislocations moving backward from where they were piled-up against the precipitate walls. From the literature and Roy et al. (2008), it is known that the pile-up length (for an applied shear stress) increases if the number of dislocations in the
! Fig. 15. Evolution of εp12 ð r Þ at different times t ¼0.2, 4.8, 8.0 and 16.0 μs for the: glide-only case (top); climb-assisted glide case (bottom).
! Fig. 16. Map of ρG ð r Þ at t ¼ 16:0 μs for the climb-assisted glide case obtained in a simulation where the extended climb kinematics is not accounted for (εp11 ¼ 0Þ.
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pile-up decreases. As the result of climb, dislocations are spreading out in the x2 direction. Since here the total number of dislocations is conserved, less dislocations will remain on an individual glide plane, and hence they will move backward to reach their new equilibrium position. So far, the full kinematics of the climb process (Eq. (35)) was incorporated in the presented results. In Fig. 16, the final spatial distribution of the GND density obtained without climb kinematics (i.e. ignoring the climb contribution to the plastic strain rate tensor) is shown. The repartition of the GNDs is quite different from the original (correct) one in Fig. 13 (t ¼ 16:0 μs) at the same instant t. Strong localization of dislocations against the wall is obtained. The omission of the εp11 contribution in the crystal plasticity framework may modify the dislocation kinetics and wrongly influence the resulting stress, strain and dislocation fields. 5. Summary and conclusion This paper presents a fully coupled extended crystal plasticity glide-climb model, in which the deformation through dislocation glide is coupled to vacancy diffusion controlled dislocation climb. Attention is given to the governing equations and the physics of the vacancy diffusion process, the driving forces acting on a dislocation causing climb and the intrinsic coupling between both. The climb law is therefore diffusion-controlled. The problem is studied at the meso-scale, where pile-ups against particles can be resolved explicitly, i.e. a scale at which it is meaningful to preserve a direct coupling between the mean vacancy flux and the dislocation climb process. A rate-dependent strain gradient crystal plasticity formulation is adopted, which accounts for the net sign of the dislocation population. The dislocation problem is described in a fully conservative manner with transport equations. Since the analysis is carried out at the meso-scale, all dislocations are assumed to be able to participate in the climbing process. The corresponding transport equations for the dislocations are updated to incorporate climb. In order to incorporate the effect of the dislocation climb on the plastic deformation tensor, the crystallographic split of the plastic strain rate tensor is extended for the climb kinematics associated to each slip system. The novel elements in this paper are:
The fully coupled continuum-based vacancy-driven climb-glide theory. The extended diffusion problem, using a local–global approach to account for the redistribution of vacancies in the continuum sense.
The extended dislocation transport equations, accounting for the vertical transport of dislocations across glide planes through climb.
The extended kinematics for the crystallographic decomposition of the plastic strain rate tensor, giving rise to a crystallographic contribution of the normal strain produced by dislocation climb. To illustrate the coupled approach and the interaction between glide and climb, two simple examples have been shown. The first concerned climbing dislocations bypassing an elastic precipitate. The second example showed how climb can relax dislocation pile-ups, by spatially redistributing the dislocations towards a low energy configuration. The theory presented is general and ready to be used in more complex 3D examples, involving multiple slip systems. The main emphasis of this paper, however, was the underlying theory, coupling and extended transport and kinematics. Further exploitation of the proposed theory and model seems well at reach. References Argon, A., Moffatt, W., 1981. Climb of extended edge dislocations. Acta Metallurgica 29, 293–299. Arzt, E., Wilkinson, D., 1986. Threshold stresses for dislocation climb over hard particles: the effect of an attractive interaction. Acta Metallurgica 34, 1893–1898. Ayas, C., Deshpande, V., Geers, M., 2012. Tensile response of passivated films with climb-assisted dislocation glide. J. Mech. Phys. Solids 60, 1626–1643. Ayas, C., van Dommelen, J., Deshpande, V., 2014. Climb-enabled discrete dislocation plasticity. J. Mech. Phys. Solids 62, 113–136. Bakó, B., Clouet, E., Dupuy, L., Blétry, M., 2011. Dislocation dynamics simulations with climb: kinetics of dislocation loop coarsening controlled by bulk diffusion. Philos. Mag. 91, 3173–3191. Bakó, B., Groma, I., Gyö rgyi, G., Zimányi, G., 2006. Dislocation patterning: the role of climb in meso-scale simulations. Comput. Mater. Sci. 38, 22–28. Bayley, C., Brekelmans, W., Geers, M., 2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. Int. J. Solids Struct. 43, 7268–7286. Beddoes, J., Wallace, W., Zhao, L., 1995. Current understanding of creep behaviour of near gamma-titanium aluminides. Int. Mater. Rev. 40, 197–217. Clouet, E., 2011. Predicting dislocation climb: classical modeling versus atomistic simulations. Phys. Rev. B 84, 092106. Danas, K., Deshpande, V., 2013. Plane-strain discrete dislocation plasticity with climb-assisted glide motion of dislocations. Model. Simulation Mater. Sci. Eng. 21, 045008. Davoudi, K., Nicola, L., Vlassak, J., 2012. Dislocation climb in two-dimensional discrete dislocation dynamics. J. Appl. Phys. 111, 103522. Dobson, P., Goodhew, P., Smallman, R., 1967. Climb kinetics of dislocation loops in aluminium. Philos. Mag. 16, 9–22. Ertürk, I., van Dommelen, J., Geers, M., 2009. Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. J. Mech. Phys. Solids 57, 1801–1814. Evers, L., Brekelmans, W., Geers, M., 2004. Non-local crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401. Gao, Y., Cocks, A., 2009. Thermodynamic variation approach for climb of an edge dislocation. Acta Mech. Solida Sin. 22, 426–435. Gao, Y., Liu, Z.L., Zhao, X.C., Zhang, Z.H., Zhuang, Z., You, X.C., 2011a. Dislocation climb model based on coupling the diffusion theory of point defects with discrete dislocation dynamics. Acta Phys. Sin. 60, 096103. Gao, Y., Zhuang, Z., Liu, Z., You, X., Zhao, X., Zhang, Z., 2011b. Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics. Int. J. Plast. 27, 1055–1071.
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